Caspian Journal of Applied Sciences Research 3(1), pp. 12-25, 2014 Journal Homepage: www.cjasr.com ISSN: 2251-9114
Distribution Pattern of Horizontal Equivalent Static Earthquake Loading for Double Layer Lattice Morteza Jamshidi1,*, Taksiah A. Majid2, Amir Darvishi1 1
Engineering faculty, Chaloos Branch, Islamic Azad University, Chaloos, 46615-397, Iran
2
School of Civil Engineering, Engineering Campus, Universiti Sains Malaysia, Nibong Tebal, 14300, Penang, Malaysia
Double layer Lattice Domes (DLLDs) are one of the most popular light weight structures selected for covering special area such as sports centers and exhibition halls. These structures are generally regarded to be safe when dealt with earthquakes. Space truss complexes consist of thousands of members and joints that make them highly redundant and stiff structures. These factors usually lead designers to ignore the dynamic analysis, especially in the preliminary design stages. The main aim of this work is to present a simple pattern for the assessment of the distribution of equivalent horizontal static earthquake loading for DLLDs to allow the designers to bypass the complexities of early stage dynamic analyses. The seismic behavior of DLLDs under twelve famous earthquake accelerograms is also presented. © 2014 Caspian Journal of Applied Sciences Research. All rights reserved.
Keywords: Lattice Domes; Double layer; Space truss; Earthquake loading
1. Introduction Space trusses are one of the famous structures that maybe used in large buildings such as exhibition halls, sports centers and airport hangers. Having extraordinary stiffness and high degree of freedom and also being light weight are some of the benefits of these types of structures. Architects are interested to use space trusses, because their production is simple and they are totally prefabricate and could easily form into different attractive geometrical surfaces. The column distance in these kinds of structures is noticeable which offers large open area for better design to architects. DLLDs are generally designed as double-layer grids with a diversity of attractive geometrical Shapes. Eekhout (2003) proposed an interesting
graph showing the rise and fall of space structures in the past decades. Space trusses are mainly consisted of thousands of steel tubular bars connected together by nodes. These connecting nodes, which play an important role in structural safety, weight, and cost of space structures, have been designed and produced in a variety of forms and systems by producers in different countries. Iffiland (1982) presented a variety of shapes and methods of assembly. Makowski (1981) studied many kinds of these nodal connection systems and Souza et al. (2003) reviewed their use in space structures. Dipaola and Prete (2003) comprehensively reported on geometrical and structural definitions of their construction systems. Space trusses are believed to attract low forces during seismic activities and can be considered to be among the least likely to suffer damage because
*
Corresponding address: Engineering faculty, Chaloos Branch, Islamic Azad University, Chaloos, 46615-397, Iran E-mail address:
[email protected] (Morteza Jamshidi)
© 2014 Caspian Journal of Applied Sciences Research; www.cjasr.com. All rights reserved.
Morteza Jamshidi; Taksiah A. Majid; Amir Darvishi / Distribution Pattern of Horizontal Equivalent Static Earthquake Loading for Double Layer Lattice 3(1), pp. 12-25, 2014
of their low weight and great stiffness when compared with other large span roofs; see Marsh (2000).
populated structures are subjected to strong earthquakes. Furthermore, no specific national or international code of practice on the design of space trusses or evaluation of their seismic behaviors exist.
Despite the earlier assumptions, recent studies show that during strong earthquakes, these structures are vulnerable to seismic failures, especially when roofs are covered with snow. Kawaguchi (1997) reported damages due to earthquake activities. Lan and Qian (1996) considered the failure of a space truss, and Kuneida et al. (2000) studied the vibrational characteristics of some existing structures. These reports and other studies carried out on seismic behavior of space structures, such as Shin et al. (2012), Sadeghi and Amani (2012), Ishikawa et al. (2000) and Ishikawa and Kato (1997), prove the importance and the need for carrying out dynamic analysis in the design of space trusses. Sadeghi (2004) investigated the dynamic characteristics of double layer barrel vaults and showed that these vaults are vulnerable to earthquakes and have a brittle behavior. In 1992, China presented some limited specifications for flat double layer space frames (JGJ 7-91). Upon these specifications, Zhang and Lan (2000) reviewed the research findings on the dynamic characteristics of space trusses.
The main aim of this work is to present a logical and simple pattern to assess the pattern of equivalent horizontal static earthquake loading distribution for Double Layer Lattice Domes (DLLDs). Further studies are currently being carried out on the effects of vertical components of earthquakes and their interaction with the horizontal components. The work presented in this paper concerns horizontal directions.
2. Method of Study 2.1. Geometrical characteristics of models Fifteen DLLDs are modeled and investigated. Linear transient analyses are carried out under horizontal seismic excitations. The results are processed to evaluate the equivalent static loading and its relative pattern of distribution over the domes. Fig. 1 shows the general geometrical properties of the models, while Table 1 presents their selected dimensions. The table shows that H, S, D, α, and R are the height, span, thickness, internal angle, and outer radius of the DLLDs, respectively (Fig. 1). These models are supported by their entire edge nodes of the inner layer at ground level by means of simple hinges. All members have tubular cross-sections with outside diameters ranging from 30 mm to 335 mm.
On the other hand, the dynamic analysis of a space truss is somehow difficult and time consuming. Thus, many structural engineers avoid this procedure using rough static loadings. These static loadings are not in any way proven to be correct. Therefore, general worries on static loading response exist, especially when important and
a. 3D view
b. Elevation
Fig. 1: General geometrical properties of models. 13
Morteza Jamshidi; Taksiah A. Majid; Amir Darvishi / Distribution Pattern of Horizontal Equivalent Static Earthquake Loading for Double Layer Lattice 3(1), pp. 12-25, 2014
members in all models. The material behavior is proposed to be elastic-perfectly plastic. However, none of the models allowed the nonlinear behavior, and only the linear part of the material behavior is contributed in analysis.
2.2. Mechanical properties of materials All models were assumed to use the same material for construction. Mild steel material, with the Young’s modulus of 200 GPa, Poisson’s ratio of 0.3, and Yield stress of 350 MPa was selected for all
Table 1: Geometrical properties of models Name of models
S (m)
α (degree)
(m)
H (m)
Name of models
S (m)
α (degree)
(m)
(m)
H (m)
(m)
M1
30.0
45.0
0.50
21.21
6.21
M9
60.0
45.0
2.00
42.43
12.43
M2
30.0
45.0
1.00
21.21
6.21
M10
60.0
45.0
2.50
42.43
12.43
M3
30.0
45.0
2.00
21.21
6.21
M11
60.0
67.5
1.50
32.47
20.05
M4
30.0
67.5
0.75
16.24
10.02
M12
60.0
67.5
2.25
32.47
20.05
M5
30.0
67.5
1.50
16.24
10.02
M13
75.0
45.0
2.25
53.03
15.53
M6
45.0
45.0
1.50
31.82
9.32
M14
75.0
67.5
2.00
40.59
25.06
M7
45.0
67.5
1.00
24.35
15.03
M15
75.0
67.5
2.25
40.59
25.06
M8
45.0
67.5
2.00
24.35
15.03
–The soil–structure interaction effect is not considered.
2.3. Loading condition In space structures, the ratio of snow load to dead load is markedly greater than in ordinary building structures. The probability of coincidence of snow and earthquake loads does not play a significant role in buildings because the snow loads are usually a negligible fraction of the total weight. On the contrary, snow loads can easily reach 2 or 3 times the self-weight of a space structure. Therefore, the combination of snow and earthquake loads in design stages is essential for consideration. The magnitude of the gravity loads in horizontal projection is considered as follows:
Table 2 lists the selected earthquakes and their characteristics. The peak ground accelerations of the selected earthquakes are different and thus, all earthquake accelerograms are scaled to 0.35 g for comparative study. 2.5. Modeling and method of analysis The space structure modeling software “FORMIAN” is used for modeling purposes. This software is based on Formex Algebra and has been developed by Nooshin and Disney (2000, 2001). FORMIAN is capable of modeling different geometrical surfaces of space structures. A typical model prepared by this software is shown in Fig. 1. In this study, the models created by FORMIAN are modified for minor adjustments using AutoCAD software (AUTODESK AG,2002). The models are then exported to SAP2000 structural analysis computer program (2009).
Dead Load = 490 Pa Snow Load = 1,373 Pa 2.4. Selected earthquake accelerograms Many criteria for choosing suitable accelerograms are investigated by researchers (Moghaddam, 2000).The following standards are considered for the selected accelerograms:
All end nodes of the inner layers are hinged to the ground. These nodes have three rotational degrees of freedom, and the three translational degrees are restrained. On the other hand, all other nodes, which are used to connect the structural members, have three translational degrees of freedom.
–The peak ground acceleration (PGA) is more than 0.3 g. –The peak acceleration to peak ground velocity (PGV) ratio is high. –The vertical to horizontal ground accelerations ratio is high. 14
Morteza Jamshidi; Taksiah A. Majid; Amir Darvishi / Distribution Pattern of Horizontal Equivalent Static Earthquake Loading for Double Layer Lattice 3(1), pp. 12-25, 2014
analyses are based upon the modal approach. At least the first 200 modes are selected to reach the desired mass participation ratios. Table 3 expresses the results of the modal analyses. A typically deformed shape is shown in Fig. 2.
3. Results and Discussion SAP2000 (2009), along with its linear time history option, is used for the dynamic analysis of the models. Overall, 180 dynamic analyses, with appropriate time steps of 0.005 s to 0.02 s and damping ratio of 2%, are carried out. The dynamic
Table 2: Selected earthquake characteristics No.
Earthquake
Date
Magnitude
PGA (g)
PGV (cm/sec)
PGD* (cm)
1
Tabas, Iran
1978/9/16
7.4
0.85
121.4
94.58
2
Chi-Chi, Taiwan
1999/9/20
7.6
0.97
107.5
18.6
3
Kobe
1995/1/16
6.9
0.82
81.3
17.68
4
Loma Prieta
1989/10/18
6.9
0.64
55.2
10.88
5
Imperial Valley
1979/10/15
6.5
0.78
45.9
14.89
6
Northridge
1994/1/17
6.7
0.94
76.6
14.95
7
San Fernando
1971/2/9
6.6
1.23
12.5
35.5
8
Duzce, Turkey
1999/11/12
7.1
0.97
36.5
5.48
9
Morgan Hill
1984/4/24
6.2
1.3
80.8
9.63
10
Bam, Iran
2003/12/26
6.5
0.8
123.51
34.6
11
Cape Mendocino
1992/4/25
7.1
0.66
89.7
29.55
12
Coalinga
1983/7/22
5.8
0.87
42.2
6.14
* :peak
ground displacement
Table 3: Modal analysis results No.
M1
M2
M3
M4
M5
M6
M7
M8
M9
M10
M11
M12
M13
M14
M15
T.M.P.*
94
94
93
94
86
83
79
84
84
84
87
89
88
90
91
T†
0.20
0.17
0.14
0.20
0.19
0.25
0.32
0.31
0.37
0.36
0.46
0.47
0.51
0.61
0.64
*: †:
Total mass participation (%) First mode period (s)
15
Morteza Jamshidi; Taksiah A. Majid; Amir Darvishi / Distribution Pattern of Horizontal Equivalent Static Earthquake Loading for Double Layer Lattice 3(1), pp. 12-25, 2014
Fig. 2: A typically deformed shape of DLLDs
The dead load consists of weights of coverings and structural elements. The live load consists of the snow load. The seismic mass of models consists of the dead load plus thirty percent (30%) of the live load.
response of the DLLDs, the seismic mass of each node ( ) is multiplied by the induced acceleration under ground motion ( ); The difference between the summation of this product for all structural nodes and the seismic base reaction (V) shows the participation of the mass matrix in the dynamic response of the lattice domes. In Eq. 2, shows the
A base shear is produced at each time step in time history analysis. The time at which the maximum base shear occurs is called the critical time. In this study, the pattern of the static equivalent seismic-induced forces over the structure is assessed at the critical time based upon the previously mentioned procedure. The dynamic equilibrium equation of a structure under seismic excitation is expressed as:
ratio of this difference to the seismic base reaction. The amount of is calculated for each model with consideration of the seismic excitation, and its average ( ) is shown in Table 4. The table shows that more than 90% of the seismic response is dependent to the structural mass and its corresponding induced acceleration. In other words, the mass and the induced acceleration of the nodes are considered as a fundamental parameter in this study, and the effect of the damping and stiffness is neglected.
(1) In Eq. 1, [M], [C], and [K] are the mass, damping, and stiffness matrices, respectively. , , and are the acceleration, velocity, and displacement vectors of the structures. is the ground acceleration vector.
(2)
According to Eq. 1, the seismic response of the structure is dependent on the structural mass, damping, and stiffness. To assess the contribution of these three mentioned matrix in the dynamic Table 4: Average of
No.
T.node is the total node number of the double lattice domes.
for each studied model
M1
M2
M3
M4
M5
M6
M7
M8
M9
M10
M11
M12
M13
M14
M15
1.43
1.2
4.1
7.3
6.3
3.7
3.5
0.8
6.
4.1
1.3
1.4
2.5
2.5
1.5
The seismic base reaction of the DLLDs is considered as the summation of the multiple mass
by the acceleration for all structural nodes as previously mentioned. For the double layer system, 16
Morteza Jamshidi; Taksiah A. Majid; Amir Darvishi / Distribution Pattern of Horizontal Equivalent Static Earthquake Loading for Double Layer Lattice 3(1), pp. 12-25, 2014
the mass of the nodes, which are located on the inner layer, is too small because it is concluded from the weight of the elements that are joined at this node, and the snow and the covering load are only an effect on the mass of the outer layer. Therefore, the greater part of the DLLDs mass is assigned to the outer layer. The ratio of the calculated seismic force for the nodes located in the outer layer to the all the nodes of the dome is presented by (Eq. 3). This amount is also
and its average ( ) is shown in Table 5. The table shows that more than 90% of the seismic force is induced to the outer layer.
(3)
where, is the number of the nodes located on the outer layer.
calculated for each model under seismic excitation
Table 5: Average of No.
for each studied model.
M1
M2
M3
M4
M5
M6
M7
M8
M9
M10
M11
M12
M13
M14
M15
93.8
94.1
93.7
93.7
93.5
93.5
95.1
91.8
91.9
91.5
92.2
91.3
90.9
91.7
91.3
On the other hand, as the surface loading of each outer layer nodes are effortlessly calculable, the distribution of the mass over the lattice domes will be easily determined. Therefore, the manner of the spread of acceleration over the domes is the only ambiguous point in the presentation of appropriate distribution pattern of horizontal equivalent static earthquake loading for DLLDs.
same horizontal rings. For each node, the ratios of the induced acceleration’s average to the maximum amount of the induced acceleration’s average on a horizontal ring are calculated and drawn against the angle between the nodes and the ground motion direction (Fig. 4). Although the verity of these acceleration ratios can be expressed as a sinuous function, earthquake is an accidental event and may accrue in all directions. Furthermore, the orientation of the future structure to the seismic excitation is an indeterminate parameter.
At the critical time, the induced acceleration’s average is calculated under all the mentioned ground motion for the nodes that are located on the 1.2
Acceleration’s Ratio
1
Ave. + δ
Angle (rad)
Ave.
0.8
Ave. - δ
0.6 0.4
Acceleration ratio diagram
0.2 0 0.00
1.57
3.14
4.71
6.28
Angle (rad)
a
b
Fig. 4: a) Distribution of induced acceleration over the horizontal ring; b) Plan view of the outer layer.
17
Morteza Jamshidi; Taksiah A. Majid; Amir Darvishi / Distribution Pattern of Horizontal Equivalent Static Earthquake Loading for Double Layer Lattice 3(1), pp. 12-25, 2014
Therefore, the average amount over the horizontal ring is preferred to be used. To show the reasonability of the use of the average amount, the standard deviation is calculated for all horizontal rings of model No. 7 (Table 6). In statistics and
probability theory, standard deviation (represented by the symbol sigma, σ) shows the variation or "dispersion" from the average. A low standard deviation indicates that the data points tend to be very close to the mean.
Table 6: Average and standard deviation of the distribution acceleration over the horizontal ring. No. of Ring
Centre
1
2
3
4
5
6
7
8
Average
---
0.99
0.97
0.94
0.89
0.85
0.81
0.74
0.63
standard deviation
---
0.00
0.01
0.04
0.06
0.09
0.13
0.15
0.10
To propose an appropriate distribution pattern of the horizontal equivalent static earthquake loading, the distribution of the induced acceleration ( ) should be extracted over the lattice domes (Fig. 3a). In Eq. 6, the horizontal ring participation acceleration ratio ( ) is defined as a ratio of the summation of the node’s acceleration that is located in the same horizontal ring ( ) to the summation of the acceleration of all outer layer nodes ( ). Fig. 5: The geometry of the outer layer. (4)
For the symmetry of the vault, only the left side of the symmetry axis is considered in the following calculation (Fig. 6).
is the number of nodes that are located over the horizontal ring. (5)
(6)
To achieve an appropriate pattern, the geometry of the vault with the acceleration distribution diagram, which is multiplied by a constant value of 0.05 (to improving the expression), is shown simultaneously. The geometry of the nodes over the vault consists of the horizontal and vertical distance from the origin, which are indexed by and consequently and are divided by the maximum value ( and ) to express the non-dimension configuration of these parameters (Eqs. 7 and 8). Fig. 5 shows that each node of the vault represents a horizontal ring, and and are also shown as the radius and elevation of the horizontal ring, respectively.
Fig. 6: Distribution of the acceleration over half of the vault.
18
Morteza Jamshidi; Taksiah A. Majid; Amir Darvishi / Distribution Pattern of Horizontal Equivalent Static Earthquake Loading for Double Layer Lattice 3(1), pp. 12-25, 2014
correlation coefficient and definition, each constant value of Eq. 9 is preferred to be expressed as a function of a geometrical property that has a noticeable correlation coefficient and , simultaneously. Some equations related to the constant values of Eq. 9 and the structural geometry with significant correlation coefficient are investigated. The equation with the maximum is suggested as the final equation. This procedure demonstrates that , , , , and is correlated to the , , , , , and , respectively (Fig. 7 and Table 9).
(7)
(8)
Fig. 6 shows the two parts of the acceleration distribution diagram. For all the studied models, Eq. 9 is used to predict both part of this diagram. The amount of Eq. 9’s constant coefficient for each model is presented in Table 7. The coefficient used for part one is indexed by 1, and index 2 is used for part two. The “Coefficient of Determination” is also calculated to assess the reliability of the fit curve equation (Kvalseth, 1985). This coefficient, which is shown by , is used in the context of statistical models whose main purpose is the prediction of future outcomes on the basis of other related information. The coefficient provides a measure of how well the future outcomes are likely to be predicted by the model.
To estimate the acceleration participation ratio of each horizontal ring, Eq. 9 should be rewritten as a function of “Z”. For Eq. 7, the following pattern is proposed to distribute the horizontal equivalent acceleration over the vault.
(11) (9)
To propose an appropriate acceleration distribution pattern, these constant coefficients should be related to the structural geometry. Eq. 10, which is presented by KARL PEARSON (2007), is used to evaluate the correlation between the structural geometry and the calculated constant coefficient. The correlation is a mathematical coefficient that determines the relation between the two parameters, which are correlated together when their value change uniformly. This correlation means that while one of the parameters increased or decreased, the other parameter also increased or decreased and their relationship can be defined by an equation. The correlation coefficient will be positive while these two parameters move in a same direction; otherwise the value must be negative if they moved in the contrary direction.
Simplifying Eq. 12 results in two different roots for , namely, the root located between zero and one valuable amount of
(10)
An example problem is presented in the Appendix.
(12)
As the left side of the symmetrical axes is considered in this study, the result of Eq. 11 will be negative. Therefore, the absolute amount of the calculation should be considered in the analysis and design of DLLDs by the structural engineers. The summation of the proposed acceleration distribution factor should be 100%; otherwise, the modification factor should be applied for scaling their summation to 100%.
Table 8 presents the amount of the correlation coefficient for all constants. Based on the
19
Morteza Jamshidi; Taksiah A. Majid; Amir Darvishi / Distribution Pattern of Horizontal Equivalent Static Earthquake Loading for Double Layer Lattice 3(1), pp. 12-25, 2014
Table 7: The amount of Eq. 9’s constant coefficient for each model.
Part 1
Part 2
M1
M2
M3
M4
M5
M6
M7
M8
1.00
1.00
1.00
1.00
1.00
1.00
1.00
0.99
0.70
0.68
0.65
0.81
0.62
0.86
0.97
0.94
0.09
0.86
-30.22
0.15
0.95
-0.08
-0.20
-1.89
0.12
0.62
-19.51
0.33
0.56
0.24
0.34
-0.87
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
0.62
0.62
0.59
0.66
0.81
0.68
0.71
0.66
0.96
0.97
0.95
0.97
-0.08
0.95
0.97
0.99
0.58
0.58
0.54
0.63
0.22
0.61
0.66
0.63
Continuation of Table 7
Part 1
Part 2
M9
M10
M11
M12
M13
M14
M15
1.00
1.00
0.99
1.00
1.00
0.99
0.99
0.90
0.91
1.00
0.97
0.92
0.95
0.97
-7.36
-6.14
-2.84
0.16
-2.92
-2.39
-2.82
-5.48
-4.68
-1.96
0.51
-2.25
-1.46
-1.82
1.00
0.99
1.00
0.99
1.00
1.00
1.00
0. 2
0.76
0.77
0.82
0.75
0.71
0.72
0.93
0.97
0.95
0.98
0.95
0.95
0.93
0.59
0.69
0.67
0.77
0.60
0.54
0.55
Table 8: Correlation coefficient between the constant values of Eq. 9 and the structural geometry parameters.
0.797
0.396
0.450
0.637
0.799
-0.478
0.828
-0.031
0.109
0.362
-0.338
-0.004
0.232
-0.810
0.159
0.325
0.083
0.391
-0.361
-0.044
0.232
-0.818
0.141
0.344
0.539
0.387
0.467
0.431
0.541
-0.2 6
0.561
-0.283
0.284
-0.230
0.059
0.323
0.155
-0.072
0.247
0.328
0.294
-0.069
0.111
0.298
0.209
-0.172
0.274
0.219
20
Morteza Jamshidi; Taksiah A. Majid; Amir Darvishi / Distribution Pattern of Horizontal Equivalent Static Earthquake Loading for Double Layer Lattice 3(1), pp. 12-25, 2014
Part 1 Part 2 1.2
0.85
R2 =0.86
1 0.8
0.75
0.6
a2
a1
R2 =0.52
0.8
0.4
0.7
0.65
0.2
0.6
0
0.55 0
10
20
30
40
50
60
0.5 15
𝑅𝛼
20
25
30
35
40
45
50
𝑅𝛼
(a)
(b) 1.2
5 0
1
R2=0.92
0.8
-10 -15
R2 =0.44
0.6
b2
b1
-5
0.4
-20 -25
0.2 0
-30
-0.2
-35 0
0.1
0.2
0.3
0
0.4
5
10
15
20
D/H (c)
0 R2=0.90
c2
c1
-5 -10 -15 -20
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
35
40
45
R2=0.42
0
-25 0.1
30
(d)
5
0
25
R/D
0.2
0.3
10
20
0.4
30
40
50
R/D
D/H Fig. 7: a) Verity of
(e) against ; b) Verity of Verity of
(f) against
against
; c) Verity of
; f) Verity of
against
against
; d) Verity of .
Table 9: Proposed equation for the constant value of Eq. 9 Part 1
Pa t 2
21
against
; e)
Morteza Jamshidi; Taksiah A. Majid; Amir Darvishi / Distribution Pattern of Horizontal Equivalent Static Earthquake Loading for Double Layer Lattice 3(1), pp. 12-25, 2014
4. Acceleration distribution curves over the definition vault practically comprise two segments, and some approximate equations are presented for estimation.
4. Conclusions The seismic behavior of DLLDs under different earthquake accelerograms was investigated, and the following results are obtained:
5. Although the acceleration distribution over the horizontal ring can be expressed as a sinuous function, earthquake is an accidental event and the orientation of the structure and seismic excitation is an indeterminate parameter for future structures. Therefore, the use of the average amount over the horizontal ring is preferred.
1. Structural mass and its corresponding induced acceleration are the main parameters in the seismic analysis of DLLDs, and more than 90% of the seismic behavior is dependent with these factors. 2. The greater part of the equivalent static earthquake loading is assigned to the outer layer because the mass of the nodes that are located on the inner layer is too small compared with the nodes located on the outer layer.
Acknowledgments Authors acknowledge the contributions from Islamic Azad university-Chaloos Branch that supported this research study.
3. As the distribution of the mass over the lattice domes is effortlessly calculable, the manner of the spread of the acceleration over the domes is the only ambiguous point in the presentation of appropriate distribution pattern of horizontal equivalent static earthquake loading for DLLDs.
No. of Ring
Appendix: Distributed constant acceleration over a DLLD with the following geometrical property (M7-Table 1):
Center
1
2
3
4
5
6
7
8
0
3.57
7.07
10.41
13.53
16.36
18.83
20.89
22.5
15.03
14.77
13.99
12.7
10.93
8.72
6.13
3.2
0
Calculation of the constant value of Eq. 9: ; The valuable amount of
For the center with
Calculation of the amount of Z_(int.):
22
;
Morteza Jamshidi; Taksiah A. Majid; Amir Darvishi / Distribution Pattern of Horizontal Equivalent Static Earthquake Loading for Double Layer Lattice 3(1), pp. 12-25, 2014
For the first horizontal ring with
;
For the seventh horizontal ring with
;
For the second horizontal ring with
;
For the eighth horizontal ring with
;
For the third horizontal ring with
;
For the fourth horizontal ring with
;
The absolute amount of the above calculations should be considered.
All of the above 1.1. (Fig. 8)
Proposed Pattern
For the fifth horizontal ring with
;
should be multiplied by
1.2
Outer Layer
Sap Results
1
𝑍_𝑗
0.8 0.6 0.4 0.2
For the sixth horizontal ring with
0
;
-1.6
-1.2
-0.8
-0.4
0
𝑋_𝑗
Fig. 8: Comparison between Sap Results and Proposed Pattern.
References Journals:
23
Morteza Jamshidi; Taksiah A. Majid; Amir Darvishi / Distribution Pattern of Horizontal Equivalent Static Earthquake Loading for Double Layer Lattice 3(1), pp. 12-25, 2014
DIPAOLA V, PRETE G (2003). Part I: Geometrical and Structural Definitions of the Construction System. International Journal of Space Structures, 18, 31-45.
International Journal of Space Structures, 27, 1522. SHIN J, LEE, K, KIM, G C, JUNG, C W & KANG J W (2012). Analytical and experimental studies on seismic behavior of double-layer barrel vault systems with different open angles. Thin-Walled Structures, 54, 113-125.
EEKHOUT M (2003). A Personal View: On the Future of Spatial Structures. International Journal of Space Structures, 18, 129-136. IFFLAND J S B (1982). Preliminary planning of steel roof space trusses. Journal of the Structural Division, 108, 2578-2590.
SOUZA A, GONCALVES R, MAIOLA C & MALITE M (2003). Theoretical analysis of the structural performance of space trusses commonly used in Brazil. International Journal of Space Structures, 18, 167-179.
ISHIKAWA K, OKUBO S, HIYAMA Y, KATO S (2000). Evaluation method for predicting dynamic collapse of double layer latticed space truss structures due to earthquake motion. International Journal of Space Structures, 15, 249-257.
ZHANG, Y. & LAN, T T (2000). Research on the Dynamic Characteristics and Seismic Response of Space Frames. International Journal of Space Structures, 15, 239-242.
KAWAGUCHI K (1997). A report on large roof structures damaged by the great Hanshin-Awaji earthquake. International Journal of Space Structures, 12, 135-148.
Diaz E, Prieto MA (2000). Bacterial promoters triggering biodegradation of aromatic pollutants.Curr.Opin.Biotech. 11(2): 467-475. Dorn E, Knackmuss HJ (1978). Chemical structure and biodegradability of halogenated aromatic compounds. Two catechol 1, 2 dioxygenases from a 3-chlorobenzoate-grown Pseudomonad.Biochem. J. 174(4): 73-84.
LAN T T, QIAN R (1996). Analysis of the failure of a space truss subjected to earthquake. 735-737. MAKOWSKI Z S (1981). Analysis, Design and Construction of Double layer Grids. Applied Science, 1-55.
Books:
MARSH C (2000). Some observations on designing double layer grids. International Journal of Space Structures, 15, 225-231.
KVALSETH T O (1985). Cautionary note about R2. American Statistician, 279-285. Conferences:
MOGHADDAM H A (2000). Seismic Behaviour of Space Structures. International Journal of Space Structures, 15, 119-135.
ISHIKAWA K, KATO S (1997). Effect of unbalanced mass distribution on vertical earthquake resistant capacity of double layer lattice domes. Proc. Of the IASS International Symposium on shell and spatial structures, Singapore.
NOOSHIN, H, DISNEY P (2000). Formex configuration processing I. International Journal of Space Structures, 15.
KUNEIDA H, MANDA T, KITAMURA K (2000). Vibrational characteristic of really existing cylindrical roof structures. Sixth Asian-Pacific conference on shell and spatial structures. Seoul, Korea.
NOOSHIN H, DISNEY P (2001). Formex configuration processing. II. International Journal of Space Structures, 16, 1-56. SADEGHI A (2004). Horizontal Earthquake Loading and Linear/Nonlinear Seismic Behaviour of Double Layer Barrel Vaults. International Journal of Space Structures, 19, 21-37.
Software: AUTODESK AG (2002). Autocad Release 11. Autodesk AG.
SADEGHI A & AMANI S (2012). Investigation of the Benefits of Application of Earthquake Action on the Design of Double Layer Barrel Vaults.
SAP2000 Advanced 14.0.0 (2009). Structural analysis program. Analysis reference manual.
24
Morteza Jamshidi; Taksiah A. Majid; Amir Darvishi / Distribution Pattern of Horizontal Equivalent Static Earthquake Loading for Double Layer Lattice 3(1), pp. 12-25, 2014
Computer
and
Structures,
Inc.
Berkley,
California, USA.
25
